3 problems

Concavity conjecture for maximal-area disk-polygons. For all $n\geq 4$,

Rounded disk-polygon billiard conjecture. Any of the shortest generalized billiard trajectories in $D(\varepsilon)$ is a $2$-periodic one for all $\varepsilon$ at most as large as…

Alexander's conjecture. Under arbitrary contraction of the center points of finitely many congruent disks in the Euclidean plane, the perimeter of the intersection of the disks can…